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Decomposing elements of a right self-injective ring

机译：分解右自射环的元素

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It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953) 358-386] and Zelinsky [Every linear transformation is sum of nonsingular ones, Proc. Amer. Math. Soc. 5 (1954) 627-630] that every linear transformation of a vector space V over a division ring D is the sum of two invertible linear transformations except when V is one-dimensional over Z_2. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra Appl. 6(2) (2007) 281-286] who proved that every element of a right self-injective ring R is the sum of two units if and only if R has no factor ring isomorphic to Z_2. In this paper we prove that if R is a right self-injective ring, then for each element a ∈ R there exists a unit u ∈ R such that both a + u and a-u are units if and only if R has no factor ring isomorphic to Z_2 or Z_3.
机译：两个Wolfson都独立地证明了这一点[所有线性变换的环的理想理论特征，Amer。 J.数学75（1953）358-386]和Zelinsky [每个线性变换都是非奇异变换的总和，Proc.Natl.Acad.Sci.USA 75：1877-410。阿米尔。数学。 Soc。参见图5（1954）627-630]，除V在Z_2上为一维时，除法环D上向量空间V的每个线性变换都是两个可逆线性变换的和。这是由Khurana和Srivastava [右自射环，其中每个元素是两个单元的和，J。Algebra Appl。 [6（2）（2007）281-286]证明了当且仅当R没有与Z_2同构的因子环时，右自射环R的每个元素都是两个单元的和。在本文中，我们证明如果R为右自射环，则对于每个元素a∈R，存在一个单位u∈R，使得当且仅当R没有因数环同构时，a + u和au均为单位到Z_2或Z_3。

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